Merging of three vortices in Eulerian and Lagrangian coordinates in a 2D flow

The left panel shows the familiar dynamics of vortices in 2D flows: they circle each other and slowly merge due to viscosity. The right panel shows the same phenomenon in Lagrangian coordinates, i.e. from the point of view of Lagrangian particles (tracers), which have been used to track the Lagrangian evolution of the vorticity field. The tracer positions were then mapped back to their point of origin.
When one transforms the Navier-Stokes equation to Lagrangian coordinates, the only term driving the dynamics stems from the viscosity of the fluid. Hence without viscosity the vorticity distribution in right panel would not change at all. The changes seen are induced by viscosity (and might be further enhanced by the resulting changes in the metric tensor). A typical feature of the dynamics in Lagrangian coordinates is that filaments of vorticity emerge which connect the vortices. This is also the case for other configurations and much higher numbers of vortices.
The visualizations are based on a simulation of the 2D Navier-Stokes equation in a periodic box of size [2*pi, 2*pi] and a simulation of Lagrangian particles (tracers) which allows it to view the dynamics in the Lagrangian coordinates.
[1] A. Daitche, Statistische und geometrische Eigenschaften turbulenter Strömungen, Master’s thesis, Institute for Theoretical Physics, University of Münster (2009).

[@ahmedalnemr3282]

Why eulerian coordinates visualize the viscosity influence however, euler equation is neglect the viscous role.

[@KillianDefaoite]

Eulerian coordinates have nothing to do with the Euler equations.